极大平面图的结构与着色理论 (4)-运算与Kempe等价类

许进

doi:10.11999/JEIT160483

极大平面图的结构与着色理论 (4)-运算与Kempe等价类

doi: 10.11999/JEIT160483

许进¹

基金项目:

国家973计划项目(2013CB329600)，国家自然科学基金(61372191, 61472012, 61472433, 61572046, 61502012, 61572492, 61572153, 61402437)

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- 文章访问数: 1508
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- 被引次数: 0
出版历程
- 收稿日期: 2016-05-11
- 修回日期: 2016-05-30
- 刊出日期: 2016-07-19

Theory on Structure and Coloring of Maximal Planar Graphs (4)-Operations and Kempe Equivalent Classes

XU Jin¹

Funds:

The National 973 Program of China (2013CB329600), The National Natural Science Foundation of China (61372191, 61472012, 61472433, 61572046, 61502012, 61572492, 61572153, 61402437)

摘要

摘要: 设G是一个k-色图，若G的所有k-着色是Kempe等价的，则称G为Kempe图。表征色数3的Kempe图特征是一尚待解决难题。该文对极大平面图的Kempe等价性进行了研究，其主要贡献是：(1)发现导致两个4-着色是Kempe等价的关键子图为2-色耳，故对2-色耳的特征进行了深入研究；(2)引入-特征图，清晰地刻画了一个图中所有4-着色之间的关联关系，并深入研究了-特征图的性质；(3)揭示了4-色非Kempe极大平面图的Kempe等价类可分为树型，圈型和循环圈型，并指出这3种类型可同时存在于一个极大平面图的4-着色集中；(4)研究了Kempe极大平面图特征，给出了该类图的多米诺递推构造法，以及两个Kempe极大平面图猜想。
- Kempe极大平面图 /
- Kempe变换 /
- -运算 /
- Kempe等价类 /
- -特征图 /
- 2-色耳
Abstract: Let G be a k-chromatic graph.G is called a Kempe graph if all k-colorings ofG are Kempe equivalent. It is an unsolved and hard problem to characterize the properties of Kempe graphs with chromatic number3. The Kempe equivalence of maximal planar graphs is addressed in this paper. Our contributions are as follows: (1) Observe and study a class of subgraphs, called 2-chromatic ears, which play a critical role in guaranteeing the Kempe equivalence between two 4-colorings; (2) Introduce and explore the properties of-characteristic graphs, which clearly characterize the relations of all 4-colorings of a graph; (3) Divide the Kempe equivalent classes of non-Kempe 4-chromatic maximal planar graphs into three classes, tree-type, cycle-type, and circular-cycle-type, and point out that all these three classes can exist simultaneously in the set of 4-colorings of one maximal planar graph; (4) Study the characteristics of Kempe maximal planar graphs, introduce a recursive domino method to construct such graphs, and propose two conjectures.
- Kempe maximal planar graph /
- Kempe transformation /
- -operation /
- Kempe equivalent class /
- -characteristic graph /
- 2-chromatic ear

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